Filling-invariants at Infinity for Manifolds of Nonpositive Curvature Noel Brady and Benson Farb

Homological invariants “at infinity” and (coarse) isoperimetric inequalities are basic tools in the study of large-scale geometry (see e.g., [Gr]). The purpose of this paper is to combine these two ideas to construct a family divk(X ), 0 ≤ k ≤ n − 2 , of geometric invariants for Hadamard manifolds X 1 . The divk(X ) are meant to give a finer measure of the spread of geodesics in X ; in fact the 0-th invariant div0(X ) is the well-known “rate of divergence of geodesics” in the Riemannian manifold X . The definition of divk(X ) goes roughly as follows (see Section 1 for the precise definitions): Find the minimum volume of a ball B needed to fill a sphere S , where S sits on the sphere S(r) of radius r in X , and the filling ball B is required to lie outside the open ball B(r)◦ in X . Then divk(X ) measures the growth of this volume as r → ∞ ; hence divk(X) is in some sense a k -dimensional isoperimetric function at infinity. We view the invariants divk(X ) in the same way as we view the standard isoperimetric inequalities (for manifolds or for groups): as basic geometric quantities to be computed. The divk(X ) are quasi-isometry invariants of X . The fundamental group π1(M ) (endowed with the word metric) of a compact Riemannian manifold is quasi-isometric to the universal cover M̃n ; hence